The class of matrices \(S(G)\)  has been of interest to many researchers recently (see
[1]}, [2]} and the references therein), and there has been considerable development of the
parameters \(M(G)\)  and \(mr(G)\) , and their positive semidefinite counterparts, see, for example, the
works [3]}, [4]}, [1]}. Furthermore, as a consequence interest has grown in connecting these
parameters to various combinatorial properties for a given graph. For example, the inverse eigenvalue problem for graphs (see [6]}) continues to receive considerable and deserved
attention, as it remains one of the most interesting unresolved issues in combinatorial
matrix theory.
